diff --git a/src/CrackNum/Main.hs b/src/CrackNum/Main.hs
index 5e40543..aca9444 100644
--- a/src/CrackNum/Main.hs
+++ b/src/CrackNum/Main.hs
@@ -563,7 +563,7 @@ encodeLane debug lanes num rm inp
 
         ef E5M2 _ = ef (FP 5 3) True -- 3 is intentional; the format ignores the sign storage, but SBV doesn't, following SMTLib
 
-        ef E4M3 _ = encodeE4M3 debug inp
+        ef E4M3 _ = encodeE4M3 debug rm inp
 
 -- | Convert certain strings to more understandable format by read
 -- If first argument is True, then we're reading using reads, i.e., haskell syntax
@@ -589,10 +589,10 @@ unrecognized inp = die [ "Input does not represent floating point number we reco
                        ]
 
 -- Encoding E4M3 is tricky, because of deviation from IEEE. So, we do a case analysis, mostly
-encodeE4M3 :: Bool -> String -> IO ()
-encodeE4M3 debug inp = case reads (fixup True inp) of
-                         [(v :: Double, "")] -> analyze v
-                         _                   -> unrecognized inp
+encodeE4M3 :: Bool -> RM -> String -> IO ()
+encodeE4M3 debug rm inp = case reads (fixup True inp) of
+                            [(v :: Double, "")] -> analyze v
+                            _                   -> unrecognized inp
  where config = z3{ crackNum = True
                   , verbose  = debug
                   }
@@ -630,30 +630,45 @@ encodeE4M3 debug inp = case reads (fixup True inp) of
          | True
          = range v
 
-       extraVals :: [(Double, String)]
-       extraVals =  [(-v, '1':s) | (v, s) <- reverse pos]
-                 ++ [( v, '0':s) | (v, s) <-         pos]
-         where pos = [ (256, "1111000")
-                     , (288, "1111001")
-                     , (320, "1111010")
-                     , (352, "1111011")
-                     , (384, "1111100")
-                     , (416, "1111101")
-                     , (448, "1111110")
+       -- This list is sorted on the first value.
+       -- Final bool is True if this value is considered "even" for rounding purposes
+       extraVals :: [(Double, String, Bool)]
+       extraVals =  [(-v, '1':s, eo) | (v, s, eo) <- reverse pos]
+                 ++ [( v, '0':s, eo) | (v, s, eo) <-         pos]
+         where pos = [ (240, "1110111", False)
+                     , (256, "1111000", True)
+                     , (288, "1111001", False)
+                     , (320, "1111010", True)
+                     , (352, "1111011", False)
+                     , (384, "1111100", True)
+                     , (416, "1111101", False)
+                     , (448, "1111110", True)
                      ]
 
        -- Pick the value we land on
        pick v = case [p | (d, p) <- dists, d == minVal] of
                   [x]    -> x
-                  [x, y] -> choose x y
+                  [x, y] -> choose v x y
                   -- The following two can't happen, but just in case:
                   []     -> error $ "encodeE4M3: Empty list of candidates for " ++ show v  -- Can't happen
                   cands  -> error $ "encodeE4M3: More than two candidates for " ++ show v ++ ": " ++ show cands
-         where dists  = [(abs (v - ev), p) | p@(ev, _) <- extraVals]
+         where dists  = [(abs (v - ev), p) | p@(ev, _, _) <- extraVals]
                minVal = minimum $ map fst dists
 
-       choose :: (Double, String) -> (Double, String) -> (Double, String)
-       choose p1 _ = p1
+       -- choose is called if we're smack in between the two values given. Then, we pick
+       -- depending on the rounding mode. Note that p1 < p2 is guaranteed here.
+       choose :: Double -> (Double, String, Bool) -> (Double, String, Bool) -> (Double, String, Bool)
+       choose v p1@(_, _, eo1) p2@(_, _, eo2) =
+           let isNegative = v < 0 || isNegativeZero v
+           in case rm of
+               RNE  -> case (eo1, eo2) of
+                         (True,  False) -> p1
+                         (False, True)  -> p2
+                         _              -> error $ "encodeE4M3: RNE can't pick between values: " ++ show (v, p1, p2)
+               RNA  -> if isNegative then p1 else p2
+               RTP  -> p2
+               RTN  -> p1
+               RTZ  -> if isNegative then p2 else p1
 
        range v
          | v < -448 || v > 448   -- Out-of-bounds becomes NaN
@@ -661,7 +676,7 @@ encodeE4M3 debug inp = case reads (fixup True inp) of
               putStrLn $ "            Note: The input value " ++ show v ++ " is out of bounds, and hence becomes NaN"
               putStrLn   "                  The representable range is [-448, 448]"
 
-         | v >= -240 || v <= 240   -- Fits into regular 4+4 format, so just decode
+         | v >= -240 && v <= 240   -- Fits into regular 4+4 format, so just decode
          = do res <- satWith config $ do x :: SFloatingPoint 4 4 <- sFloatingPoint "ENCODED"
                                          constrain $ x .== fromSDouble sRNE (literal v)
               putStrLn $ fixEncoded res
@@ -670,4 +685,3 @@ encodeE4M3 debug inp = case reads (fixup True inp) of
          -- Pick the nearest and display that
          | True
          = print $ pick v
-